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A pressure-release boundary imposes zero acoustic pressure disturbance, $\tilde{p}=0$. The incident and reflected pressure amplitudes therefore obey $1+R=0$, so the pressure reflection coefficient is $R=-1$. It contrasts with a rigid inviscid boundary, where zero normal velocity gives $R=+1$. A much lower normal impedance on the far side approximates the pressure-release condition.

A lossless intermediate section of positive real characteristic impedance $Z_M$ matches unequal real exterior characteristic impedances $Z_L,Z_R$ when $Z_M^2=Z_L{Z}_R$ and its propagation wave phase is an odd multiple of $\pi /2$. The length is an odd quarter of the internal wavelength. This principle also applies to the characteristic impedance of a stretched string.

For a uniform intermediate string of length $l$, put $\delta =k_{M}l$. Continuity of displacement and transverse force gives the displacement reflection coefficientAt real positive mechanical impedances, the denominator never vanishes. Its numerator vanishes only if $(Z_L-Z_R)\cos \delta =0$ and $(Z_M^2-Z_L{Z}_R)\sin \delta =0$. Thus unequal exterior strings require the quarter-wave condition. Equal exterior strings with a different middle string instead transmit perfectly at integer half-wavelengths; if all three mechanical impedances agree, every frequency is matched.

For a massless join between two ideal strings with a common tension $T$, require continuous transverse displacement and continuous transverse force $T{\partial }_x\Psi$. With $Z_j=\sqrt{T{\rho }_j}$ for linear mass densities ${\rho }_j$, the displacement reflection coefficient and transmission amplitude areThe reflected and transmitted mean-power fractions are $\mathcal{R}=|r{|}^2$ and $\mathcal{T}=(Z_2/Z_1)|t{|}^2$, with $\mathcal{R}+\mathcal{T}=1$. Reflection from the heavier side reverses phase; reflection from the lighter side does not. Total reflection occurs in the limits $Z_2/Z_1\to \infty$ or zero, while a finite positive impedance mismatch transmits some energy. Equal mechanical impedances remove reflection.

Two semi-infinite strings with common tension $T$ and a point mass $M$ at their join have continuous transverse displacement, $A+B=C$. With time factor $e^{i\omega t}$ and a right-moving incident factor $e^{-i{k}_{1}x}$, Newton's second law at the join gives $k_{1}T(A-B)=(k_{2}T+iM{\omega }^2)C$. Solving gives the displayed reflection coefficient and $t=C/A=2k_{1}T/[T(k_1+k_2)+iM{\omega }^2]$. Their power fractions obey $|r{|}^2+(k_2/k_1)|t{|}^2=1$. Reversing the time convention complex-conjugates the coefficients. The limits $M=0$ and $M\to \infty$ recover a massless join and a fixed boundary, respectively.

Set the lower half-space load to $q_b={\mu }_b{p}_{\beta ,b}$ using the causal vertical wavenumber branch, which excludes a wave arriving from depth infinity. Apply the uniform-layer SH impedance update upwards through every layer, keeping particle velocity and traction continuous. For upper half-space impedance $q_a$, incident velocity $V_0$ and reflected velocity $R{V}_0$, the top values are $V=V_0(1+R)$ and $T=-q_a{V}_0(1-R)$. Their ratio gives the displayed reflection coefficient, incorporating every underlying interface and multiple reflection.

A hydraulic resistance $R$ imposes $q=p/R$. Combining this with forward and backward arterial pressure waves, $p=p_{+}+p_{-}$ and $q=Y(p_{+}-p_{-})$, gives the displayed reflection coefficient. A matched load $RY=1$ has no reflected pressure wave, while a closed end has reflection coefficient one. At a network junction the combined downstream input admittance replaces $1/R$.

For a semi-infinite ideal string on $x\le 0$ with tension $T$ and linear mass density $\mu$, let a free transverse endpoint mass be $m$. Use a time-harmonic wave $y=\mathrm{Re}\{A(e^{ikx}+R{e}^{-ikx})e^{-i\omega t}\}$, where $k=\omega \sqrt{\mu /T}>0$. The endpoint boundary condition from Newton's second law is $m{y}_{tt}=-T{y}_x$. It gives the title displacement reflection coefficient, with characteristic impedance of a stretched string $Z=\sqrt{T\mu }$. Since numerator and denominator have equal moduli, all mean incident power is reflected. The endpoint kinetic energy varies within a cycle, but its mean rate of change is zero. A free mass changes the reflected phase, and does not by itself provide dissipation. The limits $m\to 0$ and $m\to \infty$ give $R=+1$ and $R=-1$ respectively at fixed positive angular frequency.

A free acoustic membrane wave is a zero of the outgoing dispersion relation $D$. At this zero, the impedance seen from one side equals the negative of that side's outgoing normal acoustic impedance. Thus the continued by analytic continuation reflection coefficient has a pole. A real guided mode is evanescent normal to the sheet; a complex root can describe a leaky mode. The pole describes nontrivial homogeneous motion with no incoming wave, rather than unbounded passive reflection at an ordinary real incidence angle.

Scattering parameters describe a linear network by relating waves incident on its ports to waves leaving them at a specified frequency. For incident-wave vector $a$ and outgoing-wave vector $b$,The entry $S_{ij}$ is outgoing wave $b_i$ divided by the incident wave $a_j$ when every other incident wave is zero. The first index is the receiving port; the second is the excited port. Keysight's primary analyzer documentation makes this order explicit and identifies $S_{11},S_{22}$ as reflection measurements and $S_{21},S_{12}$ as transmission measurements.^[ref]

For a two-port network,A matched termination at port 2 gives $a_2=0$ for a port-1 measurement. Leaving that port open generally produces a reflected incident wave instead and changes the observed input response. Port reference impedances and calibration reference planes belong to the definition of the measured network.

An original normalization calculation uses positive real reference characteristic impedance $Z_0$ and complex RMS port quantities $V,I$, with current into the network:Subtraction of their squared magnitudes givesso these wave magnitudes have squared units of power. The real-positive reference assumption is essential to this simple normalization; other reference conventions require their own definitions.

For an illustrative matched-output test with incident power $|a_1{|}^2=1\mathrm{W}$, suppose $S_{11}=0.1$ and $S_{21}=0.8e^{-i\pi /3}$. Reflected power is $0.01\mathrm{W}$ and transmitted power is $0.64\mathrm{W}$. Net power entering the device through the two ports is $0.35\mathrm{W}$. The transmission phase is $-\pi /3$; magnitude alone would omit it. These are stipulated network values, not a particular analyzer measurement.

For a one-port load, the same normalization gives reflection coefficientA $150\Omega$ resistive load at a $50\Omega$ reference gives $\Gamma =0.5$, returning one quarter of incident power. Matching requires the stated reference, not merely a small resistance.

A vector network analyzer separates reference and test receiver signals and forms the relevant complex ratios. NIST's optical-receiver metrology apparatus used electrical network and mismatch characterization alongside optical beating, illustrating why interconnecting hardware cannot always be treated as transparent.^[ref] A proposed measurement therefore records termination, reference plane, frequency and amplitude regime; an S-matrix measured for a linear operating point does not by itself characterize nonlinear large-signal behavior.

The return of an elastic wave from a contrast in material properties. At normal incidence in an isotropic medium, the conventional stress-amplitude reflection coefficient is $(Z_2-Z_1)/(Z_2+Z_1)$ for the relevant P wave or shear-wave impedance.

The sign of a reflected seismic seismic wavelet relative to an incident or reference wave under a stated recording convention. A negative normal-incidence stress-amplitude reflection coefficient reverses that polarity.

The product $Z_s=\rho V_s$ of bulk density and S wave speed. At a normal-incidence boundary it controls the corresponding stress-amplitude reflection coefficient.

An ideal short circuit fixes zero electric potential difference across a terminal pair, corresponding to zero load electrical impedance. At the end of a lossless transmission line, its voltage-wave reflection coefficient is $-1$ and the terminal voltage is zero. The available current is still limited by source and circuit impedances; real short circuits can dissipate substantial Joule heating.

On a uniform lossless transmission line, incident and reflected sinusoidal waves give voltage maxima $|V^{+}|(1+|\Gamma |)$ and minima $|V^{+}|(1-|\Gamma |)$. Their ratio is the voltage standing wave ratio, $(1+|\Gamma |)/(1-|\Gamma |)$, where $\Gamma$ is the voltage-wave reflection coefficient. A matched termination gives one, while perfect reflection gives an infinite ratio. These expressions assume a common real characteristic impedance and no propagation attenuation between the compared extrema.

A semi-infinite wave equation on a string with linear mass density $\rho$, wave speed $c$, and tension $T=\rho c^2$ has positive characteristic mechanical impedance $Z=\rho c$. At an end attached to a massless viscous damper of coefficient $\alpha >0$, displacement and transverse-force balance give the displacement reflection coefficientHere $\mathcal{A}$ is the fraction of mean incident power dissipated. A prescribed $0<\mathcal{R}<1$ gives two positive damper coefficients, $\alpha /Z=(1\pm \sqrt{\mathcal{R}})/(1\mp \sqrt{\mathcal{R}})$, with opposite reflected phases. Matching $\alpha =Z$ removes reflection.

Wave reflection returns part of an incident wave into the medium from which it arrived, as a consequence of an interface or spatial variation of material properties. An amplitude reflection coefficient compares the reflected and incident waves; its squared modulus gives their wave intensity ratio when both use the same medium and normalization. Reflected energy has not necessarily undergone wave absorption.